This study presents a radar-based algorithm for non-invasive heart rate monitoring in intensive care units (ICUs) using a 140 GHz Frequency-Modulated Continuous Wave (FMCW) radar, placed unobtrusively beneath hospital beds. Data were collected from 15 post-operative cardiac patients at Maastricht University Hospital, with an ECG device serving as the ground truth for validation. The proposed algorithm includes data preprocessing, channel selection, heart rate estimation, and post-processing, employing autocorrelation to detect rhythmic patterns and quality metrics to ensure reliable channel selection. The system achieved a mean absolute error (MAE) of 2.22 beats per minute (bpm) with 66% overall coverage, increasing to 98% during sinus rhythm periods. This approach demonstrates robust performance in challenging ICU environments by mitigating noise and motion artifacts and optimizing computational efficiency. These findings highlight the potential of radar-based systems to enhance patient care through continuous, non-invasive vital sign monitoring and validate the algorithm's effectiveness in real-world clinical scenarios.